Short answers, pulled from the story.
The United Nations officially came into existence on the 24th of October 1945. This date marked the ratification of the Charter by five permanent members of the Security Council: France, the Soviet Union, China, the United Kingdom, and the United States.
A total of 51 original states joined that year to form the organization's first roster. Fifty nations signed the Charter at the United Nations Conference on International Organization in San Francisco on the 26th of June 1945 while Poland signed later on the 15th of October 1945.
Membership is open to all states that accept the obligations contained in the present Charter and are able and willing to carry out those obligations according to the Organization's judgment. A recommendation for admission from the Security Council requires affirmative votes from at least nine of the council's fifteen members followed by approval by a two-thirds majority vote in the General Assembly.
On the 25th of October 1971, United Nations General Assembly Resolution 2758 recognized the representatives of the Government of the People's Republic of China as the only lawful representatives of China. This resolution expelled the representatives of Chiang Kai-shek from their place in the UN and effectively transferred the permanent seat from the ROC to the PRC.
Two non-member observer states participate in the UN General Assembly: the Holy See and the State of Palestine. The Sovereign Military Order of Malta also holds observer status despite not being a sovereign state while associated states like the Cook Islands and Niue remain non-members but participate in specialized agencies such as WHO and UNESCO.